Finite Group Databasebeta

← #9378

Subgroups — #9378

Order 256.

249 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3217C211yesmaximal, characteristic
2#2992C211yesmaximal, characteristic, complemented
3#1982C211yesmaximal, characteristic, complemented
4#1555C211yesmaximal, characteristic, complemented
5#1509C211yesmaximal, characteristic, complemented
6#1290C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8C2×#127C2211yescharacteristic
9#529C2211yescharacteristic
10#496C412yescomplemented
11#490C412yescomplemented
12#460—21no—
13#459—21no—
14#450—21no—
15#447—21no—
16#446—21no—
17#435—21no—
18#435—21no—
19C8×D4—21no—
20#433—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#424—21no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#102—21no—
27C2×#102—21no—
28C4×C4⋊C4C2211yescharacteristic
29#331—21no—
30#331—21no—
31#328—21no—
32#327—21no—
33C22×D4—21no—
34C22×D4—21no—
35C22×D4—21no—
36C2×SD16—22no—
37C2×D8—22no—
38C2×M16—21no—
39C22×C8—21no—
40C22×C8D411yescharacteristic
41C22×C8D411yescharacteristic
42#127—21no—
43#127—21no—
44#127—21no—
45#127C2×C412yes—
46#126—21no—
47#125C2×C411yescharacteristic
48#124C2×C411yescharacteristic
49#123—21no—
50C22⋊Q8D411yescharacteristic
51#121D411yescharacteristic
52C4×Q8D411yescharacteristic
53C4×D4—21no—
54C4×D4D411yescharacteristic
55#117—21no—
56C2×C4⋊C4—21no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D411yescharacteristic
66C4⋊C8D411yescharacteristic
67#102—22no—
68#102—22no—
69#102—42no—
70#102—42no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76#97—21no—
77#97—21no—
78#97—21no—
79#97C2×C412yes—
80C4×C8—21no—
81#95—21no—
82C24—41no—
83C2×Q8C22⋊C411yescharacteristic
84C2×D4—21no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—81no—
96C2×D4—81no—
97C2×D4—82no—
98C2×D4C22⋊C411yescharacteristic
99C22×C4—21no—
100C22×C4—41no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4#4111yescharacteristic
104SD16—82no—
105D8—82no—
106M16—81no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—81no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—41no—
122C4⋊C4C22⋊C411yescharacteristic
123C4⋊C4C22⋊C411yescharacteristic
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—21no—
129C42—21no—
130C42—81no—
131C42C22×C411yescharacteristic
132C42C2×D411yescharacteristic
133C42C2×D411yescharacteristic
134C42#4111yescharacteristic
135C23—21no—
136C23—42no—
137C23—81no—
138C23—81no—
139C23—81no—
140C23—81no—
141C23—81no—
142C23—81no—
143C23#12111yescharacteristic
144Q8—41no—
145D4—41no—
146D4—42no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—82no—
153D4—82no—
154D4—82no—
155D4—81no—
156D4—81no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C4×D411yescharacteristic
172C2×C4C4×D411yescharacteristic
173C2×C4#12011yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12311yescharacteristic
176C8—42no—
177C8—41no—
178C8—41no—
179C8—41no—
180C8—41no—
181C8—81no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—82no—
188C22—82no—
189C22—82no—
190C22—161no—
191C22—161no—
192C22—161no—
193C22—161no—
194C22#38611yescharacteristic
195C22#47111yescharacteristic
196C22#47211yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—41no—
205C4—82no—
206C4#45311yescharacteristic
207C4#45411yescharacteristic
208C2—21no—
209C2—21no—
210C2—81no—
211C2—162no—
212C2#188311yescharacteristic
213C2#190711yescharacteristic
214C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3217C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2992C2semidirect
3. #1982C2semidirect
4. #1555C2semidirect
5. #1509C2semidirect
6. #1290C2semidirect
7. #1285C2semidirect
10. #496C4semidirect
11. #490C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#32172
#29922
#19822
#15552
#15092
#12902
#12852